Deterministic long run average problems of optimal control are ''asymptotically equivalent" to infinite-dimensional linear programming problems ( LPP ) and the latter are approximated by finite dimensional LPP. The solutions of this finite dimensional LPP can be used for numerical analysis of periodic optimization problems. In the present paper we establish the convergence of controls constructed on the basis of the solution of the finite dimensional LPP to the optimal control of a periodic optimization problem. Results are illustrated with a numerical example.